Solutions to Homework #5, MAT1341A Question # 1: (2 + 2 + 2 = 6 points) Write down the general solutions of the systems with the following augmented matrices. (a) ± 1 20 1 300 10 4 ² Solution: This is already in RREF; the nonleading variables are x 2 and x 4 so we set x 2 = r and x 4 = s ; then the two rows give us the equations x 1 + 2 x 2 + x 4 = 3 (which yields x 1 = 3-2 r-s ) and x 3 = 4. Thus the general solution is x 1 = 3-2 r-s, x 2 = r, x 3 = 4 , x 4 = s for r, s ∈ R or, in vector form            3-2 r-s r 4 s     | r, s ∈ R        (b)     100 10 10 20000000 3     Solution: This is not in RREF (because the zero row is not at the bottom, for instance); but in any case the last row corresponds to a degenerate equation, meaning that this system is inconsistent. The general solution is the empty set. (c) ³0000 10 ´ Solution: This system is the equation in 5 variables: x 5 = 0. There are no conditions or constraints on any of x 1 , x 2 , x 3 or x 4 , so these are all free, and the general solution is { ( r, s, t, u, 0) | r, s, t, u ∈ R } . Note that this is the same thing you get if you follow the algorithm: this matrix is in RREF, with a leading 1 in the 5th column. Thus there are nonleading variables in the ﬁrst 4 columns, and we set their corresponding variables equal to parameters; the value of

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